Reliability of Radio-mobile systems considering fading and shadowing channels

Transcription

1 Reliability of Radio-mobile systems considering fading and shadowing channels Philippe Mary ETIS UMR 8051 CNRS, ENSEA, Univ. Cergy-Pontoise, 6 avenue du Ponceau, Cergy, France Philippe Mary 1 / 32

2 General Context Cellular mobile communications (GSM, UMTS, WLAN) Philippe Mary 2 / 32

3 Motivations To increase the quality of services (QoS) of cellular networks : Considering both short and long term effect (multipath and average power variation) In the literature Signal processing community Multipath cancellation (equalization, channel coding, smart antennas...) Network community fight again the QoS variations due to the variation of the average power Minimal research effort has been allocated to the study of both effects Our work Study of a QoS criterium considering both short and long term effects Philippe Mary 3 / 32

4 Purposes of the work Analytical performance of Radio-mobile systems in fading channels and shadowing environment ++ Allows to point out quickly the behavior of a system ++ Allows to predict behavior x x Not tractable when modelisation is more and more accurate Resource allocation considering QoS Philippe Mary 4 / 32

5 Summary 1 Channel models and assumption 2 Quick overview of SEP approximation 3 Outage considering fading and shadowing 4 Outage considering interference 5 Conclusion and further work Philippe Mary 5 / 32

6 Short term effect/long term effect Philippe Mary 6 / 32

7 Dealing with shadowing Work with outage [Conti03] : Symbol error outage (SEO) (non-ergodic channel) P s (O) = Pr (SEP P s ) The important network design criterium is the packet error outage (PEO) Philippe Mary 7 / 32

8 Fading channels model Flat fading channels The received power is : P r α shad α 2 fading α fading : Short term effect (fast fading) Nakagami-m or Rice distributed Instantaneous SNR variation γ s = α 2 fading E s /N 0 α shad : Long term effect (change after one or several packets) log-normally distributed Average SNR variation γ s = α shad E s /N 0, α shad = E ( αfading 2 ) Average SNR log-normally distributed Mean : µ db = E (10 log 10 γ s ) Standard deviation : σ db Philippe Mary 8 / 32

9 Error modelisation in fading channels Average error probability (w.r.t. short term effects) : Classic form of the average SEP : P s (E γ s ) = 0 P s (E γ s, γ s ) }{{} f (Q( γ s)) p γs γ s (γ s ) dγ s Craig91 alternative form for Gaussian function Q Allowed to derive closed-form SEP expressions in an unified way thanks to the moment generating function (MGF) of the SNR [Alouini04] : P s (E γ s ) = 1 π (M 1)π/M 0 ( M γs g ) psk sin 2 dθ θ Philippe Mary 9 / 32

10 Performance with shadowing Symbol error probability SEO : P s (O µ ) = P (P s (E γ s ) P s (E γ s ) µ ) P γs (O µ ) = P (γ s γ th µ ) Need of the SNR threshold : γ th = f (P s (E)) où f (P s (E)) = P s (E γ s ) 1 The SEO is hence : P s (O µ) = Z γth =(P s (E)) 1 0 «µdb 10 log p γs (γ s µ) dγ s = Q 10 γ s (Ps (E)) Steps Find an accurate and simple expression for SEP Inverting this expression w.r.t. the SNR Estimate the SEO σ db Philippe Mary 10 / 32

11 Invertible approximation of the SEP Asymptotic analysis [Giannakis03] : Behavior of the SEP for high SNR P s (E) (G c γ s ) G d G c Coding gain horizontal shift compared to a reference G d Diversity gain error probability slop in high SNR regime x x Not accurate at low SNR Bounds of Conti et al. [Conti03] ++ Tight bounds of the average SEP thanks to bounds on the MGF of the SNR x x One kind of channel (Nakagami-m) x x No channel coding x x No interferences Philippe Mary 11 / 32

12 Laplace method Can be used for Nakagami-m and Rice channels Integral approximation : I = h(y)e λg(y) dy y D λ R, D R The Laplace approximation of I is : 2π Ĩ = λ g (y 0 ) h(y 0)e λg(y0) and I = Ĩ { 1 + O ( λ 1)}, λ y 0 = {z} min g(y) y Philippe Mary 12 / 32

13 Scenario A simple point-to-point SISO system Goal : Estimated the probability that the SEP exceeds the threshold Philippe Mary 13 / 32

14 SEP approximation in Nakagami-m channels (1/3) The exact average SEP (M-PSK, M-QAM) can be written as [Shin04] : psk s (E γ s ) = x (γ s ) m k 1 2F 1 m, 1 «1 2 ; m + 1; x (γ s ) + k 2 F 1 2, m, 1 2 m; 3 2 ; y (γ s ), 1 g psk qam s (E γ s ) = x 1 (γ s ) m k 3 2F 1 m, 1 2 ; m + 1; x 1 (γ s ) «x 2 (γ s ) m k 4 F 1 1, m, 1; m ; x 2 (γ s ) x 1 (γ s ), 1 «2 ««Philippe Mary 14 / 32

15 SEP approximation in Nakagami-m channels (2/3) The Gauss hypergeometric function can be expressed as ( 2F 1 m, 1 ) 1 2 ; m + 1; x = B(m, 1) 1 t m 1 (1 tx) 1/2 dt, Laplace approximation by choosing [Wood03] : h(t) = B (m, 1) t 1, g(t) = {m ln t 12 } ln(1 xt) 0 Philippe Mary 15 / 32

16 SEP approximation in Nakagami-m channels (3/3) Theorem (SEP approximation) In a flat Nakagami-m fading channel, the average SEP of M-PSK/M-QAM signals is well approached by : x m P s (E γ s ) k mod ; γ s 1 x t with t = m/(m + 1), x = 1/ (1 + g mod γ s /m), g mod and k mod are modulation dependent constants. Philippe Mary 16 / 32

17 Accuracy of the approximation Philippe Mary 17 / 32

18 SEP inversion Theorem (SEP inversion) Under the same conditions as in the previous proposition, the average SNR is : ( ) γ s (Ps (E)) = c 0 (P s (E) 1 m 1 c 1 Ps (E) 1 1 ) 2m m k 1 m mod Sketch of proof. Solving in [0, 1] : (k mod ) 2 x 2m + (P s (E))2 tx (P s (E))2 = 0 Constructing the series : 8 >< >: x 0 = 0, x n+1 P = s (E) k mod «1 m `1 tx n 1 2m We can show that {x n} n N is converging towards x s Philippe Mary 18 / 32

19 SEO with shadowing Philippe Mary 19 / 32

20 SEP approximation in Rice channel Propagation with a specular component No closed-form solutions : Ps psk (E γ s ) = 1 (M 1)π/M π P qam s (E γ s ) = 4g π 0 π/2 0 ( M γs g psk sin 2 θ M γs ( g qam sin 2 θ ( avec M γs g mod ) ( sin 2 θ = (1+K)sin 2 (θ) (1+K)sin 2 (θ)+g mod γ exp s ) dθ ) dθ 4g 2 π π/4 0 ) g mod Kγ s (1+K)sin 2 (θ)+g mod γ s ( M γs g ) qam sin 2 dθ θ Philippe Mary 20 / 32

21 SEO Estimation We can show that : γ s (Ps K+K (E)) = 2 g mod W 0( πps (E)Ke K ) 1+K g mod P s = 10 2 Philippe Mary 21 / 32

22 Systems with channel coding Goal : Considering the PEO PEP inversion w.r.t. SNR Assumptions : Hard decision decoding Philippe Mary 22 / 32

23 Block codes Hamming and Golay codes Philippe Mary 23 / 32

24 PEO estimation Nakagami-m fading Packets 4600 bits, 8-PSK signal PER target of 10 1 P b = and P p (E) = 1 (1 P m (E)) N/k P m (E)(t+1)B(t+1,J t) 1 [Pm (E)(t+1)B(t+1,J t)] 1 t+1 ỹ 1 t+1 «J t 1 Philippe Mary 24 / 32

25 STBC MIMO systems The output SNR is : γ STBC = H 2 F n t R γ s γ s = E 0 /N 0 The MGF of SNR can be shown to be (without correlation) : M γstbc = (M γs ) ntnr Philippe Mary 25 / 32

26 V-BLAST MIMO systems ZF linear receiver. The substream SNR is [Gore02] : γ k = γ s [H H H] 1 kk In Rayleigh channel : Z = H H H CW nt (n r, 0, Σ nt ) We can shown that : M γk = (M γs ) nr nt+1 Philippe Mary 26 / 32

27 SEP approximation with one co-channel interference The exact SEP is obtained by averaging the conditional SEP w.r.t. the INR P s (E) = 0 P s (E γ i ) p γi (γ i ) dγ i The average SEP (M-PSK, M-QAM) with one co-channel interference in Rayleigh channel is bounded by : P s (E γ d, γ i ) 2k mod 1 + g mod γ d g mod γ d 1 + γ i γ i + 2 (1 + g mod γ d ) Philippe Mary 27 / 32

28 Accuracy of the approximation QPSK Philippe Mary 28 / 32

29 Outage probability with shadowing and co-channel interference Shadowing both SNR γ d and INR γ i are random variables We can show the average result : Theorem γ d et γ i are two random variables i.i.d. and log-normally distributed. The SEO of the desired signal with one co-channel interference is : Z P (P s (E) > Ps ) = 0 10/ log (10) σ i 2πγi e (10 log 10 γ i µ i ) 2 2σ 2 i Q µd 10 log 10 γ th (P s, γ i ) σ d «dγ i where γ th (Ps, γ i ) is the needed average SNR to reach the QoS target Ps average INR γ i. knowing the Philippe Mary 29 / 32

30 Conclusions Tools to study the performance of wireless communications in a realistic environment (Fading + Shadowing) When shadowing is considered the channel is non-ergodic The PEO is a measure of the reliability that a wireless network can offer under constraint of QoS (average PEP) Applications Resource allocation considering a target PEP with a certain outage probability Roll out prediction (WLAN, ) Philippe Mary 30 / 32

31 Further works With coding Soft decision decoding (the problem statement is different) Quantify the tradeoff Energy consumption/peo reduction Connectivity study in access network Active links : average PEP < PEP target Defined the connectivity in function of the PEP target Extension to multi-hop and cooperative networks Philippe Mary 31 / 32

32 Publications [1] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, Packet Error Outage for Coded Systems Experiencing Fading Channels and Interference in Shadowing Environment, In preparation [2] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, M. Arndt, M-ary Symbol Error Outage over Nakagami-m Fading Channels in Shadowing Environments, to appear in IEEE Transactions on Communications [3] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, M. Arndt, BPSK Bit Error Outage over Nakagami-m Fading Channels in Lognormal Shadowing Environments, IEEE Communications Letters, vol. 11, no. 7, 2007, July, pp : [4] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, M. Arndt. Estimation du taux de coupure d une liaison radio MIMO dans un canal de Nakagami avec effet de masque, GRETSI 07, Troyes, Philippe Mary 32 / 32